Practice Legendre Functions (10.2) - Partial Differential Equations - Mathematics III (PDE, Probability & Statistics)

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6 questions · about 4 min · pass mark 70%. Hints available.
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2 questions · about 1 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the Legendre differential equation?
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Look for the equation that defines Legendre functions.
Question 2.
What does 'n' represent in the Legendre equation?
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It's related to the degree of the polynomial.
3Question 3.
What is the general form of the Legendre differential equation?
- \\(\\frac{d}{dx}((1 - x^2)\\frac{dy}{dx}) + n(n + 1)y = 0\\)
- \\(\\frac{d^2y}{dx^2} + p(x)\\frac{dy}{dx} + q(x)y = 0\\)
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Check for the terms involving 'n'.
4Question 4.
True or False: Legendre polynomials are orthogonal over the interval [-1, 1].
- True
- False
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Think about the definition of orthogonality.
Question 5.
Prove the orthogonality of the first three Legendre polynomials over the interval [-1, 1].
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Set up the integrals accordingly.
Question 6.
Using Legendre polynomials, derive a series solution for a given potential in a gravitational field based on spherical symmetry.
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Think about how they approximate potentials.

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